scholarly journals Conservation Laws for Some Systems of Nonlinear Partial Differential Equations via Multiplier Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Rehana Naz
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Type System ◽  
Coupled System ◽  
Magnetized Plasma ◽  
Navier Stokes ◽  
Local Conservation ◽  
Navier Stokes Equations ◽  
Dependent Variables

The conservation laws for the integrable coupled KDV type system, complexly coupled kdv system, coupled system arising from complex-valued KDV in magnetized plasma, Ito integrable system, and Navier stokes equations of gas dynamics are computed by multipliers approach. First of all, we calculate the multipliers depending on dependent variables, independent variables, and derivatives of dependent variables up to some fixed order. The conservation laws fluxes are computed corresponding to each conserved vector. For all understudying systems, the local conservation laws are established by utilizing the multiplier approach.

scholarly journals Local Existence of Strong Solutions of a Fluid–Structure Interaction Model

2020 ◽  
Vol 22 (4) ◽  
Author(s):  
Sourav Mitra
Keyword(s):  
Stokes Equations ◽  
Local Existence ◽  
Interaction Model ◽  
Strong Solutions ◽  
Coupled System ◽  
Elastic Structure ◽  
Navier Stokes ◽  
Beam Equation ◽  
Navier Stokes Equations ◽  
System Coupling

AbstractWe are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.

scholarly journals On a mixed problem for the parabolic Lamé type operator

2015 ◽  
Vol 23 (6) ◽  
Author(s):  
Roman Puzyrev ◽  
Alexander Shlapunov
Keyword(s):  
Stokes Equations ◽  
Type System ◽  
Type Operator ◽  
Navier Stokes ◽  
Cylindrical Domain ◽  
Smooth Functions ◽  
Navier Stokes Equations ◽  
Solvability Conditions ◽  
Ill Posed ◽  
Well Posed

AbstractWe consider a boundary value problem for a Lamé type operator, which corresponds to a linearisation of the Navier–Stokes' equations for compressible flow of Newtonian fluids in the case where pressure is known. It consists of recovering a vector function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the integral representation's method we obtain a uniqueness theorem and solvability conditions for the problem. We also describe conditions, providing dense solvability of the problem.

Euler and Navier–Stokes equations in a new time-dependent helically symmetric system: derivation of the fundamental system and new conservation laws

2017 ◽  
Vol 818 ◽  
pp. 344-365 ◽  
Author(s):  
Dominik Dierkes ◽  
Martin Oberlack
Keyword(s):  
Conservation Laws ◽  
Coordinate System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Symmetric System ◽  
Present Contribution ◽  
Navier Stokes Equations ◽  
Constant Pitch ◽  
New Time

The present contribution is a significant extension of the work by Kelbin et al. (J. Fluid Mech., vol. 721, 2013, pp. 340–366) as a new time-dependent helical coordinate system has been introduced. For this, Lie symmetry methods have been employed such that the spatial dependence of the originally three independent variables is reduced by one and the remaining variables are: the cylindrical radius $r$ and the time-dependent helical variable $\unicode[STIX]{x1D709}=(z/\unicode[STIX]{x1D6FC}(t))+b\unicode[STIX]{x1D711}$, $b=\text{const.}$ and time $t$. The variables $z$ and $\unicode[STIX]{x1D711}$ are the usual cylindrical coordinates and $\unicode[STIX]{x1D6FC}(t)$ is an arbitrary function of time $t$. Assuming $\unicode[STIX]{x1D6FC}=\text{const.}$, we retain the classical helically symmetric case. Using this, and imposing helical invariance onto the equation of motion, leads to a helically symmetric system of Euler and Navier–Stokes equations with a time-dependent pitch $\unicode[STIX]{x1D6FC}(t)$, which may be varied arbitrarily and which is explicitly contained in all of the latter equations. This has been conducted both for primitive variables as well as for the vorticity formulation. Hence a significantly extended set of helically invariant flows may be considered, which may be altered by an external time-dependent strain along the axis of the helix. Finally, we sought new conservation laws which can be found from the helically invariant Euler and Navier–Stokes equations derived herein. Most of these new conservation laws are considerable extensions of existing conservation laws for helical flows at a constant pitch. Interestingly enough, certain classical conservation laws do not admit extensions in the new time-dependent coordinate system.

scholarly journals SENSITIVITY ANALYSIS FOR AN INCOMPRESSIBLE AEROELASTIC SYSTEM

2002 ◽  
Vol 12 (08) ◽  
pp. 1109-1130 ◽  
Author(s):  
MIGUEL A. FERNÁNDEZ ◽  
MARWAN MOUBACHIR
Keyword(s):  
Sensitivity Analysis ◽  
Stokes Equations ◽  
Coupled System ◽  
Navier Stokes ◽  
Aeroelastic System ◽  
Navier Stokes Equations ◽  
Lagrange Formulation ◽  
Boundary Parameter ◽  
Linearized Problem ◽  
Definition Of

This paper deals with problems arising in the sensitivity analysis for fluid-structure interaction systems. Our model consists of a fluid described by the incompressible Navier–Stokes equations interacting with a solid under large deformations. We obtain a linearized problem which allow us to compute the derivative of the state variable with respect to a given boundary parameter. We use a particular definition of the first-order correction for the perturbed state and consider a weak arbitrary Euler–Lagrange formulation for the coupled system.

Finite surface discretization for incompressible Navier-Stokes equations and coupled conservation laws

2020 ◽  
Vol 423 ◽  
pp. 109790
Author(s):  
Arpiruk Hokpunna ◽  
Takashi Misaka ◽  
Shigeru Obayashi ◽  
Somchai Wongwises ◽  
Michael Manhart
Keyword(s):  
Conservation Laws ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

Symmetry-invariant conservation laws of partial differential equations

2017 ◽  
Vol 29 (1) ◽  
pp. 78-117 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
ABDUL H. KARA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations ◽  
General Method

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, theb-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

scholarly journals Numerical investigation of an unsteady nanofluid flow with magnetic and suction effects to the moving upper plate

2020 ◽  
Vol 12 (2) ◽  
pp. 168781402090358 ◽  
Author(s):  
Muhammad Shuaib ◽  
Abbas Ali ◽  
Muhammad Altaf Khan ◽  
Aatif Ali
Keyword(s):  
Numerical Investigation ◽  
Stokes Equations ◽  
Volume Fraction ◽  
Nonlinear Partial Differential Equations ◽  
Variable Magnetic Field ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Nanofluid Flow ◽  
Navier Stokes Equations ◽  
Nicolson Scheme

The recent work provides the numerical investigation of an unsteady viscous nanofluid flow between two porous plates under the effect of variable magnetic field and suction/injection. Navier Stokes equations are modeled to study the hydrothermal properties of four different nanoparticles copper [Formula: see text], silver [Formula: see text], aluminum oxide [Formula: see text], and titanium oxide [Formula: see text]. The resultant nonlinear partial differential equations, governing the viscous fluid flow, are solved numerically using Crank–Nicolson scheme. The effect of important physical parameters such as volume fraction, magnetic strength, and porosity parameter are shown both graphically and in tabular form. It is found that due to the greatest thermal diffusivity for nanofluid [Formula: see text], comparatively the velocity increases more rapidly with the increasing value of volume fraction. Due to this effect, it is preferred to use nanofluid [Formula: see text] for transportation purposes.

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